When moving character vertex stretches (After weighting)
Try moving your character into that second position, where you can see which vertices are being left behind.  Then: 


Select the mesh.  Get into Edit Mode.
Select the vertex (make sure Armature deform is enabled to be seen in Viewport)

Hit N in the Viewport, and scroll down to the Vertex Properties and make sure Deform is selected.




From there, look for the bone (probably the name of a Forearm bone) that should be influencing that vertex, but isn't, and set the weight to 1.  Make sure to hit Normalize when your done.